A VIP ( Very Important Problem) ( for me anyway)

Consider a line of lockers, numbered consecutively, starting from 1 and continuing without bound.
(As you might have already noticed, this is not a real world problem)
Person 1 walks through and, starting with locker 1, makes sure all the lockers are closed.
Person 2 walks through, and, starting with locker 2, opens every other locker.
Person 3 walks through, starts at locker 3, and changes every third locker, opening the closed ones and closing the open ones.
Person 4 starts with locker 4 and changes the state of every fourth locker.
This continues in like manner, with person n, starting at locker n, and changing every nth locker.
(You might have noticed that this problem is not intended to practice necessary algebra skills)
This problem changed my life. I guess P.J. would call it a puzzle. I am sure most of the readers of this have heard the problem.
Oh, this is not exactly the puzzle that changed my life. That one also included a question. I think it is a better problem if the student has to decide what would be a good question. When I give this problem to students, they ask what the question is. My response is, "Oh no, did I forget to ask a question again? What do you think would be a good question? "
Someone will say something that is roughly equivalent to "Which lockers are open eventually?"
That is the question I was asked. However sometimes student's questions are better than the one I had in mind. Isn't it true that an important aspect of mathematics is deciding what questions to ask? Do we give our students a chance to do this?
But, as is my lifelong habit, I digress.
How did this problem change my life?
It was given to me in a course named "problem solving". I had been teaching for several years. I thought I was making progress but now know I had a lot to learn about how children learn. I now know I still do have a lot to learn about how children learn. I found the problem challenging and interesting, even though it has no basis in reality. I worked on it for a while, and an amazing thing happened. Once I had an answer, I became curious about why the answer ended up being so nice. I started thinking about it and without much effort I had a proof of a theorem about factors of integers, and I understood and therefore learned some interesting mathematics I did not know before.
I realized that what made this problem good was that I found it interesting, and that it created a situation where solving it helped me to understand something. It was not just an answer, but a revelation, that lead to a new understanding.
Was it important? Well, I have since used it to solve other problems, so I would say, yes.
But most importantly, I learned some significant mathematics by doing a problem. I was proud, and happy , and wanted to find other problems to work on. And I got this crazy idea that perhaps I could arrange for this sort of thing to happen in my classes for my students.
And so, for the next thirty five years, I tried to write problems that would lead my students to figuring out something for themselves.
It took a while. I tried and rejected the method of having them work a few examples in the hope they would notice the pattern I wanted them to discover. For one thing, it seemed phony but for another, they didn't learn any mathematics from working on the problem because they didn't really figure out any mathematics, they simply arrived at what I wanted them to arrive at.
I also tried worksheets. That was a total failure. For one thing, once give a worksheet, the objective of a student is to finish the worksheet. If another student has a question or observation, that interferes with the task at hand, which is to finish the worksheet. I also learned that students did not understand that they were supposed to learn something from finishing the worksheet. They were just supposed to finish it.
So, I eventually fell into a groove that I was happy with. One problem at a time. Everyone works on that problem. If you finish, compare your result with others. If you all agree, ask the next question. What if we changed the problem this way? Is there a generalization? Is there a special case? Meanwhile I walk around, rapidly, listening and looking at what is happening, waiting for the teachable moment, that moment when the entire class is ready to share their thoughts on the problem. I usually also share my thoughts at this point, especially with regard to connections and what might happen later.
Then I give them another problem to work on.
And learning happens, the students are actively engaged, and frequently I learn something I didn't know. I also learn a lot about how my students work.
Teaching by giving students a good problem is different than other ways of teaching. The teacher has to be ready for anything that comes up, and must have a plan if nothing comes up. The teacher must be willing to give students time to think of things , time to make up their mind, and time for ambiguity. The teacher must be prepared for long periods of time where everyone is wrong, and be willing to let that happen. It takes practice and a belief that eventually truth will win and incorrect reasoning will be exposed. It means giving up the role of the person who is the authority about truth. But in the end, it worked better for me than any alternative.

And, the best part is, I have a great time doing it.

And it all started with the famous locker problem, a puzzle that taught me how to become a better teacher.